Cartesian Frame Research Articles

On the orthogonality of surface wave eigenfunctions in cylindrical coordinates

SUMMARY The orthogonality of the Rayleigh wave eigenfunctions in laterally homogeneous, plane-stratified media is guaranteed by the structure of the ordinary differential equations describing elastic wave motion in cylindrical coordinates. This coupled first-order system is identical to that which characterizes 2-D plane wave propagation in a Cartesian coordinate reference frame. The orthogonality relation in 2-D can also be derived from energy considerations; however, an analogous argument in cylindrical coordinates has not hitherto been made. We derive the orthogonality relations for Rayleigh waves in 3-D from energy considerations and demonstrate that the standard (2-D) expression is, in fact, the generalization of a slightly more specific form. In addition, the cylindrical coordinate formulation permits the derivation of a functional orthogonality relation between Love and Rayleigh waves. The normalization of Love and Rayleigh wave eigenfunctions in cylindrical coordinates is shown to be related to the energy transport of a given outgoing Fourier-Bessel component across any surface which wholly encompasses the z-axis.

Spherical harmonics in a Cartesian frame

Displaying and comparing angular wave functions in both Cartesian and polar representations.

Autonomous single arm ORU changeout — strategies, control issues, and implementation

A preliminary strategy for autonomous single arm orbital replacement unit (ORU) changeout, the associated compliance and force control issues, and implementation results are described. The changeout operation is separated into subtasks, e.g. bolt-turn, compliant-grasp, and dual-pin-insertion/removal. Compliance and force control are done in Cartesian space and the selection of appropriate locations for the Cartesian control frame is discussed. The dual-pin-insertion/removal subtask requires a decision based strategy in conjunction with compliant control to prevent jamming. Actual implementation was done with a PUMA 560 6 degree of freedom manipulator although the control strategies could be implemented on any manipulator with at least 6 degrees of freedom.

Review of relationships between different strain tensors

The relationships between different Lagrangian strain tensors are summarized on the basis of an illustrative tensor diagram emphasizing the equivalence of the definitions of lattice deformation in the crystallographic and the Cartesian reference frames.

A Strapdown Inertial Algorithm Using an Earth-Fixed Cartesian Frame

An algorithm for processing strapdown inertial data in an earth-fixed Cartesian frame is developed in this paper. It is compared with the standard algorithm that uses the local-level frame and the geographic coordinate system for the model formulation. A general formulation of the modeling equations for the two approaches is given, and the linearization of the equations and the formulation of the appropriate Kalman filter are outlined. The derivation of the reference gravity model for both frames is briefly discussed, and numerically efficient formulas for the model's computation are given. In the case of the Cartesian algorithm, this leads to new formulas for all three components of the gravity vector. Real data are used to compare the two algorithms and to show that the accuracy is the same in both cases, but that the Cartesian formulation is about 30 percent more efficient.

An anatomy-based coordinate system for the description of the kinematic displacements in the human knee

The effects and interaction of the anatomical displacements in the human knee are a prerequisite to an accurate assessment and communication of the kinematic data. For the kinematic information to be used to improve diagnosis and treatment, and for better prosthetic design and installation, there must be clear, concise, and universal definitions of the displacements. In general, the displacements are defined as three translations and three rotations. In this paper, anatomic landmarks on the femur and on the tibia are used to define the locations and orientations of the six displacement axes; i.e. three translational and three rotational displacement axes. The most commonly accepted kinematic representation of the knee joint, in the literature, is a special geometry three-cylindric open chain in which the axes of the cylindric joints are defined according to the rotational displacement axes. The sequentially adjacent joint axes are assumed to not only intersect but to intersect at right-angles. The open chain permits a total of six degrees of freedom between a Cartesian reference frame attached to the femur and a Cartesian reference frame attached to the tibia. In this paper, the three rotational axes are shown to be skewed and off-set from each other, therefore, a three-cylindric open chain with skewed joint axes is proposed to measure the six displacements between the two reference frames. The authors believe that the proposed open chain is the most general to date and provides a more realistic representation of the displacements in the knee. To illustrate the significance of the reference frames on the interpretation of measured data, the anterior/posterior drawer is plotted against per cent gait cycle for three existing open chains and the proposed open chain.

A system for robot manipulation of electrical wires using vision

SUMMARYA prototype robot system for automated handling of flexible electrical wires of variable length is described. The handling process involves the selection of a single wire from a tray of many, grasping the wire close to its end with a robot manipulator, and either placing the end in a crimping press or, for tinning applications, dipping the end in a bath of molten solder. This system relies exclusively on the use of vision to identify the position and orientation of the wires prior to their being grasped by the robot end-effector. Two distinct vision algorithms are presented. The first approach utilises binary imaging techniques and involves object segmentation by thresholding followed by thinning and image analysis. An alternative general-purpose approach, based on more robust grey-scale processing techniques, is also described. This approach relies in the analysis of object boundaries generated using a dynamic contour-following algorithm. A simple Robot Control Language (RCL) is described which facilitates robot control in a Cartesian frame of reference and object description using frames (homogeneous transformations). The integration of this language with the robot vision system is detailed, and, in particular, a camera model which compensates for both photometric distortion and manipulator inaccuracies is presented. The system has been implemented using conventional computer architectures; average sensing cycle times of two and six seconds have been achieved for the grey-scale and binary vision algorithms, respectively.

Generation of optimal assumed stress expansions for hybrid-stress elements

The paper presents a simple method of systematically generating complete sets of self-equilibrating polynomial stress distributions for the development of arbitrarily shaped plane elasticity and plate bending hybrid-stress elements. Although the stresses are given in a cartesian frame, independent of the element geometry, their sequence—if truncated in pairs—preserves the invariant properties of the resulting elements. Moreover, such sequences are optimal since, in addition to equilibrium constraints, they also fulfil the compatibility condition on corresponding strains.

Six-axis bilateral control of an articulated slave manipulator using a Cartesian master manipulator

A bilateral master-slave manipulator system with different kinematics is described. In order to obtain good manoeuvrability, it is required to reduce the computational time for coordinate transformation and data communication between the master and the slave. We propose some methods for improving the performance: (1) the joints of the master manipulator are arranged along a Cartesian coordinate frame and force-controlled using a force/torque sensor in the handle, (2) the attitude of the hand or handle of the manipulators is described by quaternions, and (3) a dual CPU system connected by a serial communication line is used. A system that employs these methods realizes smooth operation and precise force feedback.

Stability Analysis of Position-Force Control Using Linearized Cartesian Space Model

The goal of this paper is to provide basic means for control theoretic analysis of position-force controlled manipulators. The rigid body dynamics of a constrained manipulator are first formulated in the joint space, and then transformed into the Cartesian constraint frame resulting in a reduced order model. Furthermore, the constraint frame model splits into separate submodels for the position and force controlled directions. A control architecture is also suggested, which according to the submodels leads to exact decoupling of the position and force controlled directions. The joint space model for the constrained manipulator is linearized, and then transformed into the constraint frame resulting also in a reduced order model. Finally, a linear, time-varying, discrete-time state-space model is formulated on the basis of the linearized Cartesian space model. The state-space model is of the standard form, and therefore allows the usage of basic control theoretic methods for analyzing the local stability of a position-force controlled manipulator.

Determining the representative composition of a set of sandstone samples

AbstractThe description of sandstone samples, defined by three components, is introduced through non-parametric sample density description. The composition of a particular sample defines a direction from the origin of the Cartesian reference frame. Radial difference between individual samples then provides the measure of their similarity. A sample set can be viewed as a bundle of compositional axes with a corresponding density distribution on both the ternary diagram (or unit constant sum plane) and the positive octant of the unit sphere. Because radial differences are disproportionately displayed on the ternary diagram, radial density calculations on the sphere are to be preferred. The mode of the sample set, the implied centre of greatest data concentration, is considered to be the most meaningful representative value, as the component joint mean can lie outside the data field with certain data configurations. In addition, the component joint mean cannot provide a useful representative composition for samples drawn from a bimodal distribution. Sample density distribution not only provides a data defined approach to describe sample dispersion, but also furnishes an indication of sample sufficiency in displaying the degree to which modes are supported by surrounding observations.

Invariance and the uncertainty principle in EPR spectroscopy of crystals and liquids

The ambiguities in the spin Hamiltonian formalism describing electron paramagnetic resonance spectra are discussed in terms of the uncertainty principle of quantum mechanics. The observables in the spin Hamiltonian can be defined as the invariants under transformation in the spinor basis of effective spins, or the invariants under rotation in the Cartesian frame, if the spectra are recorded in crystals or isotropic liquids, respectively. The separability of scalar, antisymmetric and tensorial components of magnetic interactions in liquids, as well as the non-separability in crystals is considered as the uncertainty in the measurements of angle and angular momentum.

Angles measure compositional differences

The geometry of petrological sample space shows that a rock sample, defined by a particular composition or proportion of components, is uniquely described by a radial direction (= compositional axis) from the origin of a Cartesian reference frame where amounts of each component are measured along the axes. The original measured amounts determine the direction of the compositional axis defining a particular mixture of components. Normalization (i.e., expressing amounts as proportions of the whole) merely provides radial projection of the sample onto a proportions frame (three components = the ternary diagram) to permit a form of comparison of different compositions. Such comparisons become inconsistent when traditional parametric approaches are used because, as the constant sum problem has long revealed, numbers expressing proportions or percentages are not amenable to standard statistical methods. In addition, because of the disproportionate display of radial difference over the ternary diagram, it cannot be used as a basis for reliable calculations. Recognition of the fundamental radial character of compositional data reveals that they should, most logically, be analyzed on the sphere. Not only does such an approach bypass previously recognized problems in statistical treatment of compositional data, but also it opens the way for the application to petrological data of a whole class of established parametric and nonparametric statistical methods.

An algorithmic approach to coordinate transformation for robotic manipulators

Coordinate transformation is one of the most important issues in robotic manipulator control. Robot tasks are naturally specified in work space coordinates, usually a Cartesian frame, while control...

Robust compliant motion for manipulators, part I: The fundamental concepts of compliant motion

A method for the design of controllers of constrained manipulators in the presence of model uncertainties is developed. The controller must carry out fine maneuvers when the manipulator is not constrained, and compliant motion, with or without interaction-force measurement, when the manipulator is constrained. At the same time stability must be preserved if bounded uncertainties are allowed in modelling the manipulators. Stability of the manipulator and environment as a whole and the preservation of stability in the face of changes are two fundamental issues that have been considered in the design method. A set of practical design specifications in the frequency domain is presented that is meaningful from the standpoint of control theory and assures the desired compliant motion in the Cartesian coordinate frame and stability in the presence of bounded uncertainties. This approach also assures the global stability of the manipulator and its environment. The consequence of inexact achievement of performance specifications on stability is also specified. Part I concerns the fundamentals of compliant motion, while Part II is devoted to the controller design method.

An assessment of numerical solutions of the compressible Navier-Stokes equations

Nomenclature e = internal energy e = e{ + ¥211-u e^ = specific internal energy / =force F, G, H = vector fluxes J = Jacobian of coordinate transformation k = molecular heat conductivity n — outward normal P = pressure Pr = Prandtl number Prt = turbulent Prandtl number q =heat exchange Q = heat release t = time T = temperature u, v, w = velocity components in Cartesian frame x, y, z = Cartesian coordinates 7 = ratio of specific heats e = eddy viscosity coefficient X = bulk viscosity coefficient fji = molecular viscosity coefficient £, 17, f = transformed coordinates p = density T = stress tensor

Perceptual organization as nested control.

It is argued here that perceptual organization is the structuring of environmental stimuli into nested structures of control. This view is derived from our proposal that the perceptual system forces any stimulus set to have the algebraic structure G of a machine, and that G is given a specific factorization sequence, G = G1 X G2 X ... X Gn, induced by dynamical systems criteria which the organism imposes on the environment. An investigation into these factorization sequences reveals that any such sequence is split into two subsequences, one structuring the way in which the stimulus set can change under external action, and the other describing the way in which the set is perceived as internally generated. An examination of the nested control structure of the latter yields a theory of grouping. The external and internal sequences are shown to be strongly related to each other in that the symmetry axes of the internal sequence are eigenspaces of the most stable subgroup factors of the external sequence. It is claimed that cartesian reference frames are subsequences of the full sequences. Using these principles, a unified theory is offered of several apparently quite separate perceptual areas; e.g. Marr-Nishihara shape perception, gestalt grouping, the orientation-and-form problem, and motion perception. The final claim is that planning hierarchies have the same dynamically-structured algebraic sequences and therefore that the study of perceptual organization should, in a very deep sense, be formally equivalent to the study of planning.

Dielectric currents in the low‐latitude boundary layer and geomagnetic tail

We consider a collisionless plasma with E (the electric field) parallel to the y‐axis of a cartesian frame of reference and B (the magnetic induction) parallel to the z‐axis. When there is a gradient, orthogonal to B, of the quantity ϵE², where ϵ = 1 + 4πρm c²/B², where ρm is the plasma mass density and c the speed of light, there is driven a (dielectric) current (j) given in the following two cases, by urn:x-wiley:00948276:media:grl2299:grl2299-math-0001 (i) When ▽(ϵE²) is parallel to E, j is in the direction of E × B, and (ii) when ▽(ϵE²) is in the direction E × B, j is in the direction of E. In the former case it follows that, in a steady state, urn:x-wiley:00948276:media:grl2299:grl2299-math-0002 along a line in the y‐direction, and in the latter case urn:x-wiley:00948276:media:grl2299:grl2299-math-0003 on a line in the x‐direction where p⊥ is the pressure perpendicular to B. The currents related to ϵE² are geophysically significant in the region of interaction of the solar wind and geomagnetic field in the low latitude boundary layer and geomagnetic tail. The last written equation enables a realistic estimate of the cross‐tail electric field to be made.

Kinematic equations for industrial manipulators

A method for developing the kinematic equations of motion for a six degree‐of‐freedom manipulator is presented, which can be applied to most commercially available robots. Cartesian coordinate frames are assigned to each link so that the number of transcendental and arithmetic operations needed to transform from coordinates in one frame to any other frame is minimised.

The Intrinsic Geometrical and Physical Properties of Plane Steady Gas Flow

The main purpose of the present paper is to establish a relation between the intrinsic geometrical and physical properties of plane steady gas flow, i.e., between the geometry of the flow pattern on one side and the flow variables on the other. This is accomplished by complete integration of the governing fluid dynamics equations in an a priori unknown system of curvilinear coordinates. The complete solution is presented by explicit expressions for the metric tensor components associated with the curvilinear system in terms of the flow Mach number and the Bernoulli constant along the streamlines. The curvilinear system relative to a Cartesian reference frame is a posteriors determined by quadratures. Several examples including subsonic, sonic, supersonic, rotational, and irrotational flow are discussed to illuminate the special technique and its potential for determining flow patterns that result from particular geometrical and/or physical specifications.